Subgroups — C₃×#81782 #11874285

Order 1152.

1225 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#912	C₂	1	4	yes	maximal, complemented
2	C₃×#5716	C₂	1	2	yes	maximal, complemented
3	C₃²×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5155	C₂	1	4	yes	maximal, complemented
5	C₃×#5076	C₂	1	2	yes	maximal, complemented
6	C₃×#4699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3147	—	3	1	no	maximal
9	#81782	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#237	C₂²	1	4	yes	—
11	C₂×C₃²×D₈	C₂²	1	4	yes	—
12	C₃²×#127	C₂²	1	2	yes	—
13	C₃×#938	—	2	4	no	—
14	C₃×#936	C₂²	1	4	yes	—
15	C₃×D₄×Dic₃	—	2	4	no	—
16	C₂×C₃×#214	C₂²	1	8	yes	—
17	C₃×#912	—	2	16	no	—
18	C₂×C₃×Dic₁₂	C₂²	1	4	yes	—
19	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
20	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
21	C₃×#869	C₂²	1	2	yes	—
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₈×Dic₃	C₂²	1	2	yes	—
24	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃×#135	—	3	4	no	—
26	C₃×#532	—	3	2	no	—
27	C₂×#912	C₂×C₃	1	4	yes	complemented
28	#5716	C₂×C₃	1	2	yes	complemented
29	C₃×#495	—	3	4	no	—
30	C₃×#494	—	3	1	no	—
31	C₃×#493	—	2	1	no	—
32	C₃×#493	D₃	1	1	yes	characteristic, complemented
33	C₃×#493	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#492	—	3	2	no	—
35	C₂×C₃×C₄×C₈	—	3	1	no	—
36	#5155	C₂×C₃	1	4	yes	complemented
37	#5076	C₂×C₃	1	2	yes	complemented
38	#4699	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃²×D₄	—	2	4	no	—
41	C₂×C₃²×D₄	C₂³	1	4	yes	—
42	C₂×C₃×#67	—	2	4	no	—
43	C₂²×C₃×Dic₃	—	2	4	no	—
44	C₃×#237	—	2	16	no	—
45	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
46	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
47	C₂×C₃×Dic₆	C₂³	1	4	yes	—
48	C₃²×D₈	—	2	8	no	—
49	C₂×C₃²×C₈	C₂³	1	2	yes	—
50	C₃²×C₄²	C₂³	1	1	yes	characteristic
51	C₃×#217	—	2	8	no	—
52	C₃×#214	—	2	16	no	—
53	C₃×C₄⋊Dic₃	—	2	4	no	—
54	C₃×C₄×Dic₃	D₄	1	2	yes	—
55	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
57	C₃×Dic₁₂	—	2	8	no	—
58	C₃×D₃×C₈	D₄	1	8	yes	complemented
59	#3147	—	3	1	no	Sylow
60	C₂×C₃×#41	—	3	4	no	—
61	C₂×#237	C₂²×C₃	1	4	yes	—
62	C₃×#135	—	6	16	no	—
63	C₂×C₃×Q₁₆	—	3	4	no	—
64	C₂×C₃×SD₁₆	—	3	8	no	—
65	C₂×C₃×D₈	—	2	4	no	—
66	C₂×C₃×D₈	C₂×D₃	1	4	yes	—
67	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
68	C₂²×C₃×C₈	—	3	2	no	—
69	C₃×C₄⋊Q₈	—	3	2	no	—
70	C₃×#127	—	2	2	no	—
71	C₃×#127	C₂×D₃	1	2	yes	—
72	C₃×#127	C₂²×C₃	1	2	yes	—
73	C₃×#124	—	3	4	no	—
74	C₃×#121	—	6	4	no	—
75	C₃×C₄×D₄	—	6	4	no	—
76	C₂×C₃×C₄²	—	3	1	no	—
77	#938	—	2	4	no	—
78	#936	C₂²×C₃	1	4	yes	—
79	D₄×Dic₃	—	2	4	no	—
80	C₂×#214	C₂²×C₃	1	8	yes	—
81	#912	—	2	16	no	—
82	C₂×Dic₁₂	C₂²×C₃	1	4	yes	—
83	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
84	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
85	#869	C₂²×C₃	1	2	yes	—
86	C₃×C₄×C₈	—	2	1	no	—
87	C₃×C₄×C₈	—	3	2	no	—
88	C₃×C₄×C₈	—	3	1	no	—
89	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
90	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
91	C₈×Dic₃	C₂²×C₃	1	2	yes	—
92	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
93	C₂³×C₃²	—	2	4	no	—
94	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
95	C₃²×D₄	—	2	8	no	—
96	C₃²×D₄	—	4	8	no	—
97	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
98	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
99	C₃×#67	—	4	8	no	—
100	C₂×C₃×Dic₃	—	2	4	no	—
101	C₂×C₃×Dic₃	—	4	8	no	—
102	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
103	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
104	C₃×C₄×D₃	—	2	4	no	—
105	C₃×C₄×D₃	C₂×D₄	1	4	yes	—
106	C₃×Dic₆	—	2	8	no	—
107	C₃²×C₈	C₂×D₄	1	4	yes	—
108	C₃×C₃⋊C₈	C₂×D₄	1	4	yes	—
109	C₂×#135	—	3	4	no	—
110	#532	—	3	2	no	—
111	#495	—	3	4	no	—
112	#494	—	3	1	no	—
113	#493	C₃×D₃	1	1	yes	characteristic, complemented
114	#492	—	3	2	no	—
115	C₂×C₄×C₈	—	3	1	no	—
116	C₃×#41	—	6	16	no	—
117	C₂×C₃×Q₈	—	3	4	no	—
118	C₂×C₃×D₄	—	2	4	no	—
119	C₂×C₃×D₄	—	2	4	no	—
120	C₂×C₃×D₄	—	2	4	no	—
121	C₂×C₃×D₄	—	4	4	no	—
122	C₂×C₃×D₄	—	6	4	no	—
123	C₂×C₃×D₄	C₂²×D₃	1	4	yes	—
124	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
125	C₂²×C₃×C₄	—	3	1	no	—
126	C₂²×C₃×C₄	—	3	2	no	—
127	C₂²×C₃×C₄	—	6	4	no	—
128	C₂×#67	—	2	4	no	—
129	C₂²×Dic₃	—	2	4	no	—
130	#237	—	2	16	no	—
131	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
132	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
133	C₂×Dic₆	C₂³×C₃	1	4	yes	—
134	C₃×Q₁₆	—	6	8	no	—
135	C₃×SD₁₆	—	6	16	no	—
136	C₃×D₈	—	2	8	no	—
137	C₃×D₈	—	2	8	no	—
138	C₃×D₈	—	4	8	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	3	2	no	—
141	C₂×C₃×C₈	—	3	8	no	—
142	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
143	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
144	C₃×C₄⋊C₄	—	6	4	no	—
145	C₃×C₂²⋊C₄	—	6	8	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	3	2	no	—
148	C₃×C₄²	—	3	1	no	—
149	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
150	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
151	#217	—	2	8	no	—
152	#214	—	2	16	no	—
153	C₄⋊Dic₃	—	2	4	no	—
154	C₄×Dic₃	C₃×D₄	1	2	yes	—
155	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
157	Dic₁₂	—	2	8	no	—
158	D₃×C₈	C₃×D₄	1	8	yes	complemented
159	C₂²×C₃²	—	4	4	no	—
160	C₂²×C₃²	—	4	8	no	—
161	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₃×D₃	—	2	2	no	—
163	C₂×C₃×D₃	#127	1	2	yes	—
164	C₃²×C₄	#135	1	4	yes	complemented
165	C₃²×C₄	C₂²×D₄	1	2	yes	—
166	C₃×Dic₃	—	2	2	no	—
167	C₃×Dic₃	—	4	4	no	—
168	C₃×Dic₃	#127	1	2	yes	—
169	C₂×#41	—	3	4	no	—
170	#135	—	6	16	no	—
171	C₂×Q₁₆	—	3	4	no	—
172	C₂×SD₁₆	—	3	8	no	—
173	C₂×D₈	C₂×C₃×D₃	1	4	yes	—
174	C₂²×C₈	—	3	2	no	—
175	C₄⋊Q₈	—	3	2	no	—
176	#127	C₂×C₃×D₃	1	2	yes	—
177	#124	—	3	4	no	—
178	#121	—	6	4	no	—
179	C₄×D₄	—	6	4	no	—
180	C₂×C₄²	—	3	1	no	—
181	C₄×C₈	—	3	2	no	—
182	C₄×C₈	—	3	1	no	—
183	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
184	C₂³×C₃	—	2	4	no	—
185	C₂³×C₃	—	2	4	no	—
186	C₂³×C₃	—	3	1	no	—
187	C₂³×C₃	—	4	4	no	—
188	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
189	C₃×Q₈	—	6	8	no	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	—	2	8	no	—
192	C₃×D₄	—	4	8	no	—
193	C₃×D₄	—	4	8	no	—
194	C₃×D₄	—	4	8	no	—
195	C₃×D₄	—	8	8	no	—
196	C₃×D₄	—	12	8	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	3	1	no	—
200	C₂×C₃×C₄	—	3	4	no	—
201	C₂×C₃×C₄	—	3	2	no	—
202	C₂×C₃×C₄	—	6	4	no	—
203	C₂×C₃×C₄	—	6	4	no	—
204	C₂×C₃×C₄	—	12	8	no	—
205	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
207	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
209	#67	—	4	8	no	—
210	C₂×Dic₃	—	2	4	no	—
211	C₂×Dic₃	—	4	8	no	—
212	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
213	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
214	C₄×D₃	—	2	4	no	—
215	C₄×D₃	C₂×C₃×D₄	1	4	yes	—
216	Dic₆	—	2	8	no	—
217	C₃×C₈	—	2	4	no	—
218	C₃×C₈	—	3	4	no	—
219	C₃×C₈	D₃×D₄	1	4	yes	—
220	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
221	C₃⋊C₈	C₂×C₃×D₄	1	4	yes	—
222	C₂×C₃²	—	8	4	no	—
223	C₂×C₃²	C₂×#127	1	1	yes	characteristic
224	C₂×C₃²	C₂×#135	1	2	yes	—
225	C₃×D₃	—	2	2	no	—
226	#41	—	6	16	no	—
227	C₂×Q₈	—	3	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	6	4	no	—
230	C₂×D₄	C₂²×C₃×D₃	1	4	yes	—
231	C₂²×C₄	—	3	1	no	—
232	C₂²×C₄	—	3	2	no	—
233	C₂²×C₄	—	6	4	no	—
234	Q₁₆	—	6	8	no	—
235	SD₁₆	—	6	16	no	—
236	D₈	—	2	8	no	—
237	C₂×C₈	—	3	2	no	—
238	C₂×C₈	—	3	8	no	—
239	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
240	C₄⋊C₄	—	6	4	no	—
241	C₂²⋊C₄	—	6	8	no	—
242	C₄²	—	3	1	no	—
243	C₄²	—	3	2	no	—
244	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	3	2	no	—
247	C₂²×C₃	—	4	8	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	8	no	—
250	C₂²×C₃	—	4	4	no	—
251	C₂²×C₃	—	6	2	no	—
252	C₂²×C₃	—	8	4	no	—
253	C₂²×C₃	—	8	8	no	—
254	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
255	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
256	C₂×D₃	—	2	2	no	—
257	C₂×D₃	C₃×#127	1	2	yes	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	3	2	no	—
261	C₃×C₄	—	6	2	no	—
262	C₃×C₄	—	12	4	no	—
263	C₃×C₄	#912	1	4	yes	complemented
264	C₃×C₄	C₃×#135	1	4	yes	complemented
265	C₃×C₄	C₂×D₃×D₄	1	2	yes	—
266	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
267	Dic₃	—	2	2	no	—
268	Dic₃	—	4	4	no	—
269	Dic₃	C₃×#127	1	2	yes	—
270	C₃²	#3147	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	2	4	no	—
272	C₂³	—	3	1	no	—
273	Q₈	—	6	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	4	8	no	—
276	D₄	—	12	8	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	4	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	6	4	no	—
281	C₂×C₄	—	6	4	no	—
282	C₂×C₄	—	12	8	no	—
283	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
284	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
285	C₈	—	3	4	no	—
286	C₈	C₃×D₃×D₄	1	4	yes	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	6	2	no	—
290	C₂×C₃	—	8	4	no	—
291	C₂×C₃	—	8	4	no	—
292	C₂×C₃	—	16	4	no	—
293	C₂×C₃	D₃×#127	1	1	yes	characteristic
294	C₂×C₃	C₂×#912	1	2	yes	—
295	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
296	C₂×C₃	C₂×C₃×#135	1	2	yes	—
297	D₃	—	2	2	no	—
298	C₂²	—	3	2	no	—
299	C₂²	—	4	8	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	6	2	no	—
302	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
303	C₄	—	3	2	no	—
304	C₄	—	6	2	no	—
305	C₄	—	12	4	no	—
306	C₄	C₃×#912	1	4	yes	complemented
307	C₄	C₂×C₃×D₃×D₄	1	2	yes	—
308	C₃	—	2	1	no	—
309	C₃	#81782	1	1	yes	characteristic, direct product factor
310	C₃	C₃×#3147	1	1	yes	characteristic, complemented
311	C₂	—	6	2	no	—
312	C₂	—	8	4	no	—
313	C₂	C₃×D₃×#127	1	1	yes	characteristic
314	C₂	C₂×C₃×#912	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#493	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₃×#493	D₃
hypocenter	C₃	C₃×#3147
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#493

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×#912	C₂	semidirect
2. C₃×#5716	C₂	semidirect
3. C₃²×#493	C₂	semidirect
4. C₃×#5155	C₂	semidirect
5. C₃×#5076	C₂	semidirect
6. C₃×#4699	C₂	semidirect
7. C₃×C₄×D₃×C₈	C₂	semidirect
9. #81782	C₃	direct (N × H)
27. C₂×#912	C₂×C₃	semidirect
28. #5716	C₂×C₃	semidirect
32. C₃×#493	D₃	semidirect
33. C₃×#493	C₂×C₃	semidirect
36. #5155	C₂×C₃	semidirect
37. #5076	C₂×C₃	semidirect
38. #4699	C₂×C₃	semidirect
39. C₄×D₃×C₈	C₂×C₃	semidirect
58. C₃×D₃×C₈	D₄	semidirect
113. #493	C₃×D₃	semidirect
158. D₃×C₈	C₃×D₄	semidirect
164. C₃²×C₄	#135	semidirect
263. C₃×C₄	#912	semidirect
264. C₃×C₄	C₃×#135	semidirect
270. C₃²	#3147	semidirect
306. C₄	C₃×#912	semidirect
309. C₃	#81782	direct (N × H)
310. C₃	C₃×#3147	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#912	2
C₃×#5716	2
C₃²×#493	2
C₃×#5155	2
C₃×#5076	2
C₃×#4699	2
C₃×C₄×D₃×C₈	2
C₃×#3147	3
#81782	3

Subgroups — C3×#81782 #11874285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#81782 #11874285